GMAT Data Sufficiency-Is \(\frac{7^7}{7^x}\) an integer?

Question: Is \(\frac{7^7}{7^x}\)  an integer?

1. \(0\leq{x}\leq7\)​
2. \(|x|=x^2\)

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“Is  \(\frac{7^7}{7^x}\) an integer?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Answer:

Approach Solution 1

\(\frac{7^7}{7^x}=7^{7-x}\), so as long as x is an integer and x \(\leq\)7, the expression is an integer.

Now considering S(1), we have  \(0\leq{x}\leq7\)

If x is an integer from this range, then \(\frac{7^7}{7^x}\) will be an integer but if x is a fraction from this range, then \(\frac{7^7}{7^x}\) will NOT be an integer.

Now we will see the S(2), we have \(|x|=x^2\)

Do squaring on both sides, we will get:

\(x^2=x^4\)

\(x^2(x^2-1)=0\)

\(x^2(x-1)(x+1)=0\)

X = 0, 1, -1

For each of these value \(\frac{7^7}{7^x}\) IS an integer. So, we have an YES answer.

Correct option: B

Approach Solution 2

From \(|x|=x^2\) , it does not follow |x| = x

• If x is negative, we will have:

\(-x=x^2\) (Notice here that –x there is NOT negative, its negative positive)

x (x+1) = 0

x = 0 or x = -1

x = -1

• If x is positive or o, then we’d get:

\(x=x^2\)

x (x-1) = 0

x = 0 or x = 1

So, x = -1, 0, 1

Correct option: B

Suggested GMAT Quant Questions

• GMAT Data Sufficiency – If |m + 4| = 2, what is the value of m?
• GMAT Data Sufficiency – If a > b, how much greater than b is a?
• GMAT Data Sufficiency – What is the radius of the circle above with center O?
• GMAT Data Sufficiency – P and Q are prime numbers less than 70. What is the units digit of P*Q?
• GMAT Data Sufficiency – Is 500 the average (arithmetic mean) score on the GMAT?
• GMAT Data Sufficiency – If 4x = 5y = 10z, what is the value of x + y + z ? (1) x - y = 6 (2) y + z = 36
• GMAT Data Sufficiency – Can the Positive Integer p be Expressed as the Product of Two Integers
• GMAT Data Sufficiency – A certain truck uses 1/12 +kv^2 gallons of fuel per mile when its speed is v miles per hour.
• GMAT Data Sufficiency – What is the value of x? (1) x^2 – 5 x + 6 = 0 (2) x > 0
• GMAT Data Sufficiency – What is the value of x? 1) x^2 + x + 10 = 16 2) x = 4y^4+2y^2+2
• GMAT Data Sufficiency – A terminating decimal is defined as a decimal that has a finite number of nonzero digits
• GMAT Data Sufficiency – A rectangular solid has length, width, and height of L cm, W cm, and H cm, respectively
• GMAT Data Sufficiency – Triangle ABC is right angled at B. BD, the median to hypotenuse AC, is 5 units
• GMAT Data Sufficiency – If each side of parallelogram P has length 1, what is the area of P ?
• GMAT Data Sufficiency – Is the positive integer n a multiple of 24?
• GMAT Data Sufficiency – Is 1/(a - b) > b - a ?
• GMAT Data Sufficiency – If J≠0, What Is The Value Of J?
• GMAT Data Sufficiency – Paint X Costs $15 per Gallon, and Paint Y Costs $20 per Gallon
• GMAT Data Sufficiency – What is the total value of Company H's stock?
• GMAT Data Sufficiency – Is PQ > 1?